Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes

TL;DR

This paper introduces a minimal set of edge modes for Chern-Simons theory that preserve topological invariance, leading to a novel factorization map and insights into 3D gravity and quantum groups.

Contribution

It proposes a minimal edge mode construction for Chern-Simons theory that maintains topological invariance and applies to 3D gravity, connecting to quantum group degrees of freedom.

Findings

Minimal edge modes can be interpreted as particle degrees of freedom on a quantum group.

The minimal factorization map uniquely applies to 3D gravity with SL(2,R) gauge group.

The approach aligns with previous ideas linking black hole entropy to topological entanglement entropy.

Abstract

One approach to analyzing entanglement in a gauge theory is embedding it into a factorized theory with edge modes on the entangling boundary. For topological quantum field theories (TQFT), this naturally leads to factorizing a TQFT by adding local edge modes associated with the corresponding CFT. In this work, we instead construct a minimal set of edge modes compatible with the topological invariance of Chern-Simons theory. This leads us to propose a minimal factorization map. These minimal edge modes can be interpreted as the degrees of freedom of a particle on a quantum group. Of particular interest is three-dimensional gravity as a Chern-Simons theory with gauge group SL$(2,\mathbb{R}) \times$ SL$(2,\mathbb{R})$. Our minimal factorization proposal uniquely gives rise to quantum group edge modes factorizing the bulk state space of 3d gravity. This agrees with earlier proposals that relate the Bekenstein-Hawking entropy in 3d gravity to topological entanglement entropy.